Random ergodic theorems and real cocycles |
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Authors: | Mariusz Lemańczyk Emmanuel Lesigne François Parreau Dalibor Volný Maté Wierdl |
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Institution: | 1. Department of Mathematics and Computer Science, Nicholas Copernicus University, ul. Chopina 12/18, 87-100, Toruń, Poland 2. Laboratoire de Mathématiques et Physique Théorique, UMR 6083 CNRS, Université Fran?ois Rabelais, Parc de Grandmont, 37200, Tours, France 3. Laboratoire d'Analyse, Géométrie et Applications, UMR 7539, Institut Galilée, Université Paris 13, 99, av. J.B. Clément, 93430, Villetaneuse, France 4. Laboratoire Rapha?l Salem, UMR 6085 CNRS, Université de Rouen, 76821, Mont Saint-Aignan, France 5. Department of Mathematical Sciences, University of Memphis, 38152, Memphis, TN, USA
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Abstract: | We study mean convergence of ergodic averages
associated to a measure-preserving transformation or flow τ along the random sequence of times κ
n
(ω) given by the Birkhoff sums of a measurable functionF for an ergodic measure-preserving transformationT.
We prove that the sequence (k
n(ω)) is almost surely universally good for the mean ergodic theorem, i.e., that, for almost every, ω, the averages (*) converge
for every choice of τ, if and only if the “cocycle”F satisfies a cohomological condition, equivalent to saying that the eigenvalue group of the “associated flow” ofF is countable. We show that this condition holds in many natural situations.
When no assumption is made onF, the random sequence (k
n(ω)) is almost surely universally good for the mean ergodic theorem on the class of mildly mixing transformations τ. However,
for any aperiodic transformationT, we are able to construct an integrable functionF for which the sequence (k
n(ω)) is not almost surely universally good for the class of weakly mixing transformations. |
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